Student's t-statistic

(noun)

a ratio of the departure of an estimated parameter from its notional value and its standard error

Related Terms

Examples of Student's t-statistic in the following topics:

  • Multivariate Testing

    • Hotelling's $T$-square statistic allows for the testing of hypotheses on multiple (often correlated) measures within the same sample.
    • A generalization of Student's $t$-statistic, called Hotelling's $T$-square statistic, allows for the testing of hypotheses on multiple (often correlated) measures within the same sample.
    • Hotelling's $T^2$ statistic follows a $T^2$ distribution.
    • Hotelling's $T$-squared distribution is important because it arises as the distribution of a set of statistics which are natural generalizations of the statistics underlying Student's $t$-distribution.
    • The test statistic is defined as:

  • The t-Test

    • A t-test is any statistical hypothesis test in which the test statistic follows a Student's t-distribution if the null hypothesis is supported.
    • A t-test is any statistical hypothesis test in which the test statistic follows a Student's t-distribution if the null hypothesis is supported.
    • When the scaling term is unknown and is replaced by an estimate based on the data, the test statistic (under certain conditions) follows a Student's t-distribution.
    • All such tests are usually called Student's t-tests, though strictly speaking that name should only be used if the variances of the two populations are also assumed to be equal.
    • Writing under the pseudonym "Student", Gosset published his work on the t-test in 1908.

  • Comparing Two Sample Averages

    • Student's t-test is used in order to compare two independent sample means.
    • The result is a t-score test statistic.
    • A t-test is any statistical hypothesis test in which the test statistic follows Student's t distribution, as shown in , if the null hypothesis is supported.
    • If using Student's original definition of the t-test, the two populations being compared should have the same variance.
    • If the sample sizes in the two groups being compared are equal, Student's original t-test is highly robust to the presence of unequal variances.

  • The t-Distribution

    • Fisher, who called the distribution "Student's distribution" and referred to the value as $t$.
    • Student's $t$-distribution with $\nu$ degrees of freedom can be defined as the distribution of the random variable $T$:
    • This distribution is important in studies of the power of Student's $t$-test.
    • Student's $t$-distribution arises in a variety of statistical estimation problems where the goal is to estimate an unknown parameter, such as a mean value, in a setting where the data are observed with additive errors.
    • In any situation where this statistic is a linear function of the data, divided by the usual estimate of the standard deviation, the resulting quantity can be rescaled and centered to follow Student's $t$-distribution.

  • t-Test for One Sample

    • The formula for the $t$-statistic $T$ for a one-sample test is as follows:
    • Under $H_0$ the statistic $T$ will follow a Student's distribution with $19$ degrees of freedom: $T\sim \tau \cdot (20-1)$.
    • Compute the observed value $t$ of the test statistic $T$, by entering the values, as follows:
    • Determine the so-called $p$-value of the value $t$ of the test statistic $T$.
    • The Student's distribution gives $T\left( 19 \right) =1.729$ at probabilities $0.95$ and degrees of freedom $19$.

  • Confidence Interval, Single Population Mean, Standard Deviation Unknown, Student's-t

    • This problem led him to "discover" what is called the Student's-t distribution.
    • For each sample size n, there is a different Student's-t distribution.
    • A probability table for the Student's-t distribution can also be used.
    • A Student's-t table (See the Table of Contents 15.
    • The notation for the Student's-t distribution is (using T as the random variable) is

  • Summary of Formulas

    • Use the Student's-t Distribution with degrees of freedom df = n − 1.
    • $EBM = t_{\frac{\alpha }{2}} \cdot \frac{s}{\sqrt{n}}$

  • Distribution Needed for Hypothesis Testing

    • Perform tests of a population mean using a normal distribution or a student's-t distribution.
    • (Remember, use a student's-t distribution when the population standard deviation is unknown and the distribution of the sample mean is approximately normal. ) In this chapter we perform tests of a population proportion using a normal distribution (usually n is large or the sample size is large).

  • Student Learning Outcomes

    • By the end of this chapter, the student should be able to:

  • Assumptions

    • Assumptions of a $t$-test depend on the population being studied and on how the data are sampled.
    • Most $t$-test statistics have the form $t=\frac{Z}{s}$, where $Z$ and $s$ are functions of the data.
    • If using Student's original definition of the $t$-test, the two populations being compared should have the same variance (testable using the $F$-test or assessable graphically using a Q-Q plot).
    • If the sample sizes in the two groups being compared are equal, Student's original $t$-test is highly robust to the presence of unequal variances.
    • Welch's $t$-test is insensitive to equality of the variances regardless of whether the sample sizes are similar.